Talk:Cantor–Dedekind axiom

This is not an "axiom" in the usual sense of mathematics, though it might be an axiom in some broader natural-language sense. I'd like to see a source for the claim that either Cantor or Dedekind formulated, proposed, or indeed even believed it.--Trovatore 1 July 2005 05:56 (UTC)

• I second this. Is says that "real numbers are order-isomorphic to the linear continuum of geometry". Reals are what their axioms say they are. "Linear continuum", according to the wiki, is either the real line or a linearly ordered set. In the former sense, the statement is trivially true. In the latter sense, "real numbers are order-isomorphic to a linearly ordered set in geometry", or may be "to a linear ordering in geometry". I find this statement misleading since reals are, in fact, linearly ordered and we do not need a thesis (an ordinary language axiom) to see that all of the results of the classical geometry remain theorems in analytic geometry. It is of notice, though, that the classical geometry can also be attempted with other DLOs, like rationals, may be? melikamp (talk) 17:56, 15 September 2009 (UTC)
  • My impression is that modern mathematics does not have the concept of a platonic geometric "line", but classical geometry did. And if you add this axiom to classical (Euclidean?) geometry, then it turns out that your plane is the real plane, and your lines are real lines, and you can apply the methods of analytic geometry. I'm not sure if this actually makes sense. However, I'm going to delete the sentence that says that this axiom is actually a theorem, as it isn't at all clear what theory it claims it's a theorem of. It seems to be meaningless in standard modern math -- as you pointed out, there is no relevant notion of a "linear continuum" which isn't defined to be the real numbers. And I can't begin to understand what it would mean to prove it in classical geometry -- I don't think you can use Euclid's postulates to construct the real numbers and then prove that they're order-isomorphic to the points on the line. I couldn't find the strength of Euclid's axioms from a cursory skim of the article on Euclidean geometry, but there must be things which can be proven in analytic geometry which can't be proven from Euclid's postulates, right? 68.61.127.138 (talk) 07:42, 31 December 2010 (UTC)
• It is quite clear that this does not have any meaning in the mathematical sense, especially from the viewpoint of the pure mathematician. However, it is a very fundamental assumption that certainly should not be left unstated. If anything, the name is certainly woeful --- it is more of a postulate than an axiom, and even then, more useful to the philosophy, meta-physics and meta-mathematics than mathematics itself. Please do not delete this. The fact is that, formally and philosophically, you cannot prove that the real numbers as constructed in the mathematical/axiomatic sense has anything to do with geometrical lines, even though many axioms were picked in its favour (without considering the difficulty between physical and theoretical Euclidean geometrical lines). This postulate of concern demands that there be a one-to-one correspondence between the mathematical model of Reals and the mathematical/physical model/object of lines. I shall cite the historical precedent that, when Newton wrote down his model of the world in the Principia, he specifically wrote down the assumption that "mathematical time" is the same, and flows the same, for all observers. It is precisely this assumption that is revamped by Einstein. It is even more amazing that Newton would have considered this important enough to state, given how fundamental it is to all of the computational needs of his theory. No matter how fundamental or in-consequent an assumption would seem to practitioners now, explicitly stating them down and recognising their existence is always better than hiding them. Finally, the technical details is that, even though the Reals are Dedekind-complete by axiom, it is not proven that there is a order-isomorphic map from the Reals to the line, especially the surjectivity. 94.194.100.255 (talk) 01:13, 16 October 2011 (UTC)
  • While these discussions are old, for other readers I think it would be good to add: the surjectivity part is a consequence of taking Archimedes' postulate to be true: that given two linear magnitudes, one can multiply one magnitude a finite number of times and and have it surpass the other in length. Euclid seemed to accept this postulate as he stated it in his Definition V.4 and treated it as though it were true for all the magnitudes he considered thereafter, although why he left it as a "definition" instead of stating it as an axiom outright is not something I would know why. He did something similar for circles, where he defined a diameter as "bisecting" a circle and taking this as a truth, even though it is impossible to prove it from the other axioms and it's stuck in the "definition" instead of the axioms. Personally, I think that Euclid's lines were the constructible numbers, thus much smaller than the real numbers, not much bigger. This might explain why Euclid would like the Archimedean postulate, because you are not going to be able to construct an infinitesimal from a finite line segment using a finite number of compass-and-straightedge operations (keeping in mind, of course, that Euclid was not thinking of "numbers" but rather lines and other shapes of different sizes and lengths). mike4ty4 (talk) 05:20, 3 April 2016 (UTC)

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Fixed the worst problems (which seemed to come directly from MathWorld--N.B. MathWorld is a very unreliable source!!!!) But I still think maybe it should be VfD'd. Comments solicited--Trovatore 7 July 2005 02:09 (UTC)

Continuum hypothesis

Is this just the Continuum hypothesis? --MarSch 16:00, 13 December 2006 (UTC)

No, it has nothing to do with the continuum hypothesis. As far as I can tell it's an assertion of a connection between a geometrical concept (the line) and an arithmetical one (the reals). Which seems hardly necessary to mention, given that the geometrical line is the motivation for the reals in the first place. But I suppose you could posit an axiomatic framework in which both the line and the reals are primitive concepts, or defined from distinct primitive concepts, and you'd then need some axiom like this to connect them. --Trovatore 17:21, 13 December 2006 (UTC)

It's a theorem

The Cantor–Dedekind axiom is actually a theorem. It is possible to construct a complete Archimedean field on a line in Euclidean Geometry in a fashion that preseres order. Since all complete Archimedean fields are isomorphic, the constructed field is isomorphic to the real line.

Of course, you need the right axioms for the reals and for Euclidean Geometry; the axioms in The Elements are actually not sufficient to prove the results Euclid claimed. That as pretty much cleaned up in the 19th and early 20th centuries. Shmuel (Seymour J.) Metz Username:Chatul (talk) 19:58, 30 September 2018 (UTC)

I'm adding references showing the construction of the Reals in Projective Geometry and proving that the Reals are unique up to isomorphism.

Notes

References